Subject: Calculus

# Infinite Series

An infinite series is defined as the sum of infinitely many terms, such as

1+ \frac{1}{3}+ \frac{1}{5}+\frac{1}{7}+ ... +\frac{1}{2n-1}+ ... = \sum_{n=0}^{\infty}\frac{1}{2n-1}

Some irrational numbers like pi, \pi = 3.14159265358979..., can be expanded into a form of an infinite series as follows

\pi =3+ \frac{1}{10}+ \frac{4}{10^2}+\frac{1}{10^3}+\frac{5}{10^4}+\frac{9}{10^5}+\frac{2}{10^6}+\frac{6}{10^7}+\frac{5}{10^8}+\frac{3}{10^9}+...

This in turn tells us that \pi is an infinite sum of fractions of this form.

## Importance:

Basic yet complex functions such as trigonometric and inverse trigonometric functions, exponential and logarithmic function, and rational functions can be understood well through infinite series. These functions can be simplified by expressing them in terms of infinite series in such a way that all of these functions can be approximated by polynomials, the simplest kind of function. These simpler functions can then be used to approximate more complex functions. Also, infinite series can show the close relationship between functions which seem to be quite different at first just like exponential and trigonometric functions.

## Convergent and Divergent Infinite Series

Definition. Let a_1, a_2, a_3 ,..., a_n be an infinite sequence of real numbers. The infinite series \sum_{n \geq 1}^{\infty}a_n is defined to be

\sum_{n \geq 1}^{\infty}a_n=\lim_{N \to \infty}\sum_{n}^{N}a_n
.

If the limit of such series exist in R, then we say that \sum_{n \geq 1}^{\infty}a_n is convergent, else, if the limit does not exist or is \pm infty then it is divergent. Take note that we are not actually adding up all terms of the infinite series but we are adding up a finite number of terms and see their behavior as this terms approaches infinity. By this, we then define an infinite series as the limit of its sequence of partial sums.

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